A209242 The largest fixed value (neither happy nor sad) in base n.

8, 1, 18, 1, 45, 52, 50, 1, 72, 125, 160, 1, 128, 1, 261, 260, 200, 1, 425, 405, 490, 1, 338, 1, 657, 628, 450, 848, 936, 845, 1000, 832, 648, 1, 1233, 1377, 800, 1, 1450, 1445, 1813, 1341, 1058, 1856, 2125, 1844, 1250, 1525, 1352, 2205, 2560, 1, 2873, 1, 3200

Offset

3,1

Comments

A number is a fixed value if it is the sum of its own squared digits. Such values >1 are the only numbers that are neither happy (A007770) nor unhappy (A031177) in that base.

The number of fixed values in base B (A193583) is equal to one less than the number of divisors of (1+B^2) (Beardon, 1998, Theorem 3.1).

No fixed point has more than 2 digits in base B, and the two-digit number a+bB must satisfy the condition that (2a-1)^2+(2b-B)^2=1+B^2 (Beardon, 1998, Theorem 2.5). Since there are a finite number of ways to express 1+B^2 as the sum of two squares (A002654), this limits the search space.

Because fixed points have a maximum value of B^2-1 in base B, there are a large number of solutions near perfect squares, x^2. Surprisingly, there are also a large number of points near "half-squares", (x+.5)^2. See "Ulam spiral" in the links.

Links

Table of n, a(n) for n=3..57.

Christian N. K. Anderson, All fixed values in base n for n=3..10000

Christian N. K. Anderson, Ulam spiral of maximum fixed values in base n for=3..1000

Alan F. Beardon, Sums of Squares of Digits, The Mathematical Gazette, 82(1998), 379-388.

Example

a(7)=45 because in base 7, 45 is 63 and 6^2+3^2=45. The other fixed values in base 7 are 32, 25, 10 and (as always) 1.

Prog

(R) #ya=number of fixed points, yb=values of those fixed points
library(gmp); ya=rep(0, 200); yb=vector("list", 200)
for(B in 3:200) {
  w=1+as.bigz(B)^2
  ya[B]=prod(table(as.numeric(factorize(w)))+1)-1
  x=1; y=0; fixpt=c()
  if(ya[B]>1) {
    while(2*x^2<w) {
      if(issquare((y=as.numeric(w-x^2)))) {
        y=sqrt(y)
        av=(1+rep(c(-1, -1, 1, 1), 2)*rep(c(x, y), e=4))/2
        bv=(B+rep(c(-1, 1), 4)*rep(c(y, x), e=4))/2
        ix=av>=0 & av<B & bv>=0 & bv<B & !(av==0 & bv==0) & isint(av)
        fixpt=c(fixpt, (av+B*bv)[ix])
      }
      x=x+1
    }
  } else fixpt=1
  yb[[B]]=sort(unique(fixpt))
}
sapply(yb, max)
(Python)
from sympy.ntheory.digits import digits
def ssd(n, b): return sum(d**2 for d in digits(n, b)[1:])
def a(n):
    m = n**2 - 1
    while m != ssd(m, n): m -= 1
    return m
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Aug 01 2021

Crossrefs

Cf. A007770, A031177.

Cf. A193583.

Sequence in context: A013615 A359628 A369404 * A103884 A103883 A379447

Adjacent sequences: A209239 A209240 A209241 * A209243 A209244 A209245

Keyword

nonn,base

Author

Christian N. K. Anderson, Apr 22 2013

Extensions

Program improved and sequence extended by Christian N. K. Anderson, Apr 25 2013.

Status

approved